PAPERSPhysics on a Branched Knotted Spacetime Manifold
This paper reproduces the dynamics of quantum mechanics with a four-dimensional spacetime
manifold that is branched and embedded in a six-dimensional Minkowski space. Elementary
fermions are represented by knots in the manifold, and these knots have the properties of the
familiar particles. We derive a continuous model that approximates the behavior of the manifold's discrete branches. The model produces dynamics on the manifold that corresponds to the
gravitational, strong, and electroweak interactions.

Knot Physics: Neutrino Helicity
We use the assumptions of knot physics to prove that a collection of interacting neutrinos and
antineutrinos maximize their quantum probability when all neutrinos are of the same helicity and
all antineutrinos are of the opposite helicity. In a previous paper we showed that the geometry of
gravity spontaneously breaks symmetry. We show here that the geometry of gravity couples the
neutrino linear momentum to its quantum phase. Likewise, the quantum phase of an interacting
neutrino couples to its spin angular momentum. Therefore, the symmetry breaking of gravity
couples the linear momentum of an interacting neutrino to its spin angular momentum, producing
consistent helicity.

Knot Physics: Deriving the Fine Structure Constant
Knot physics describes the geometry of particles and fields. In a previous paper we described the topology and geometry of an electron. From the geometry of an electron we can construct a mathematical model relating its charge to its spin angular momentum. From experimental data, the spin angular momentum is &hbar;/2. Therefore the mathematical model provides a comparison of electron charge to Planck's constant, which gives the fine structure constant α. We find that using only electromagnetic momentum to derive the fine structure constant predicts a value for α-1 that is about two orders of magnitude too small. However, the equations of knot physics imply that the electromagnetic field cusp must be compensated by a geometric field cusp. The geometric cusp is the source of a geometric field. The geometric field has momentum that is significantly larger than the momentum from the electromagnetic field. The angular momentum of the two fields together predicts a fine structure constant of α-1≈136.85. Compared to the actual value of α-1≈ 137.04, the error is 0.13%. Including the effects of virtual particles may reduce the error further.

Knot Physics: Entanglement and Locality
We describe entanglement and locality in knot physics. In knot physics, spacetime is a branched manifold. The quantum information of a system is encoded in the branches of the manifold. We show how that quantum information can persist despite the continual recombination of the branches of the manifold. We also note that the quantum collapse of state of the branches is non-local. That non-locality allows for non-local effects of entanglement without additional assumptions. We apply this description to the EPR paradox.

Knot Physics: Dark Energy
We describe features of cosmology in knot physics that have the characteristics of dark energy. Knot physics assumes that spacetime is a branched 4-manifold embedded in a Minkowski 6-space. The cosmology of an embedded spacetime manifold is described by the expansion and contraction of the manifold in the embedding space. We show that the motion of the manifold in the embedding space contributes to the redshift of photons on the manifold. In this way, the embedded manifold model provides an alternative explanation for the redshift data that has been used as evidence for dark energy.

Knot Physics: Dark Matter
We describe dark matter in knot physics. Knot physics assumes that spacetime is a branched 4-manifold embedded in a Minkowski 6-space. The theory has three metrics. The Minkowski space has the standard Minkowski metric ημν. The second metric, ημν, is just the restriction of ημν to the spacetime manifold. In a previous work, we showed how mass and energy affect the curvature of ημν, reproducing results of general relativity. The third metric, gμν, is used to constrain the branches of the spacetime manifold. In this paper, we derive an approximate relationship between ημν and gμν. The relationship implies ημν can have non-zero Ricci curvature without a massive source particle. We show how this result has many of the characteristics of dark matter.